The Octonion Loop

[Steve]

Quaternions can be considered as a sort of hypercomplex number by replacing the real coefficients in a complex number with other complex numbers. The octonions similarly replace the real coefficients of a quaternion with yet another depth of complex numbers. They give rise to  an 8 dimensional algebra. There is one major difference. Although the complex numbers and quaternions form a group, the octonions are not because octonions fail the associative law in some cases, ie. a(bc) does not equal (ab)c.  Octonions fall under the category of "loops".

As it turns out, the associative law is not used in formulae needed in the recursion relation so there is no problem. Using octonions in the recursion, O_n+1 = O_n² + O₀, leads to interesting images that don't occur in the Mandelbrot set.

Three types of images are shown here. The first is harsh with abrupt edges. The second is very soft and delicate. Both of these examples are fractals, but you have to zoom in on the occasional small details to see the next level. Other examples are shown at http://www.insidetheoutbox.net/recmaps/recmap-index.php?genre=oct



This image nothing but long curved thin lines. If you zoom into the dark areas, and increase the number of iterations, you will see that they are comprised of more thin threads. If you zoom in on the knotted up region at the bottom center. You will find different types of conjunctions of lines similar to the large picture. This goes on to the limit of resolution.

[Krumel]

Very nice, and now go on to sedenions

[Steve]

You must have read my mind. 

[Calcyman]

Steve, I was considering an octonionic fractal a few days ago. You must have read my mind.

Instead of sedenions (which are boring, anyway), why don't you try using the Greiss algebra, with 196884 dimensions?

There are plenty of other spaces, as well. I don't think anyone has attempted fractals on the hyperbolic plane.

[Steve]

I think you are joking. The number of floating point ops per pixel goes up roughly as the square of the dimension.  The octonion has about 95 flops per pixel per iteration. The Sedonion will have about 318. For image sizes that I am using, a 16-D space takes two or three minutes.  The Griess algebra would take well over a year per picture.

I have done a number of smaller groups, which are listed in the sidebar at http://www.insidetheoutbox.net/recmaps/ If you have any good suggestions on groups with less than 100 elements I would consider doing them.

[Calcyman]

I think you are joking.

Yes, the 196884-dimensional fractal was never intended to be rendered on a computer.


Well, there's a nice 1D abelian group based on elliptic curves:

http://en.wikipedia.org/wiki/Elliptic_curve#The_group_law

You could adjoin two of them to form a 2D algebra, similar to how the complex numbers are formed from two real numbers. You'll have to define multiplication yourself, though.